Of the following sets of four quantum numbers \(\left\\{n, l, m_{l}, m_{s}\right\\}\), identify the ones that are forbidden for an electron in an atom and explain why they are invalid: (a) \(\left\\{2,2,-1,+\frac{1}{2}\right\\} ;\) (b) \(\left\\{6,6,0,+\frac{1}{2}\right\\} ;\) (c) \(\left\\{5,4,+5,+\frac{1}{2}\right\\}\)

The principal quantum number, denoted by the symbol \(n\), plays a pivotal role in quantum mechanics, particularly when describing the energy levels of an electron within an atom. It is integral for determining the overall size and energy of an electron's orbit. The principal quantum number can be any positive integer, starting from 1 upward. The significance of \(n\) lies in its determination of the shell in which the electron resides, with \(n=1\) indicating the innermost shell closest to the nucleus, and higher values of \(n\) corresponding to shells farther away.The principal quantum number also influences the potential energy of an electron; the larger the value of \(n\), the higher the energy level, and the further away the electron is likely to be from the nucleus. This results in larger orbitals and a higher probability of finding the electron at a greater distance from the nucleus. It is this quantization of the electron's energy levels that makes the concept of principal quantum number a fundamental aspect of quantum mechanics.

The angular momentum quantum number, represented by \(l\), is a quantum number that defines the shape of an electron's orbital. It can take on any integer value from 0 up to \(n-1\), with \(n\) being the principal quantum number associated with the electron. Each value of \(l\) correlates with a specific subshell, which are designated by letter codes: \(l=0\) corresponds to the s-subshell, \(l=1\) to the p-subshell, \(l=2\) to the d-subshell, and \(l=3\) to the f-subshell, and so on.For example, if an electron occupies a shell with \(n=3\), the possible values for \(l\) would be 0, 1, or 2 (which refer to the s, p, and d subshells, respectively). The angular momentum quantum number is directly related to the angular momentum of an electron within the atom and thus helps to provide a deeper understanding of the orbital structure and the complex nature of electron behavior within the atom.

The magnetic quantum number, denoted \(m_l\), specifies the orientation of an electron's orbital in space relative to an external magnetic field. It is the third quantum number in the set that characterizes the unique state of an electron and is dependent on the angular momentum quantum number \(l\). For any given value of \(l\), the \(m_l\) can have integer values ranging from \( -l \) to \( l\), including zero. Thus, there are \(2l+1\) possible values for \(m_l\) for each value of \(l\).This quantum number arises because an electron's orbital can take on various orientations, each described by different values of \(m_l\). The number of orientations and thus the possible values of \(m_l\) grow as the value of \(l\) increases. This is a reflection of the more complex geometries and spatial configurations that are possible with orbitals of higher angular momentum.

The concept of electron spin refers to an intrinsic form of angular momentum carried by electrons, which is independent of their orbital angular momentum. It is described by the spin quantum number, \(m_s\), which can have only two possible values: \(+1/2\) or \( -1/2\). These values represent two different orientations of the electron's spin: one aligned with an external magnetic field (commonly referred to as 'spin-up') and the other opposed to it ('spin-down').Electron spin is a fundamental property that has profound implications in fields such as chemistry and solid-state physics, notably in the explanation of the Pauli exclusion principle, which dictates that no two electrons in the same atom can have identical sets of quantum numbers. This property impacts the arrangement of electrons in an atom and plays a crucial role in the formation of chemical bonds and the behavior of materials in the presence of magnetic fields.

The Framework of Atoms

Quantum mechanics is the branch of physics that deals with the behavior of matter and light on the atomic and subatomic scale. It embodies a completely different set of principles compared to classical mechanics, with the core idea being that energy, momentum, angular momentum, and other quantities of a bound system are quantized. Quantum mechanics has revolutionized our understanding of the atomic and molecular world - introducing concepts like wave-particle duality, the uncertainty principle, and quantum entanglement.

Underpinning Technology

Contemporary technologies such as computers, smartphones, and magnetic resonance imaging (MRI) scanners, are all based on the principles of quantum mechanics. Furthermore, research into quantum computers shows the potential for this theory to further revolutionize technology by dramatically increasing computational power.